# American Institute of Mathematical Sciences

2015, 2015(special): 1034-1040. doi: 10.3934/proc.2015.1034

## Recurrence of multi-dimensional diffusion processes in Brownian environments

 1 Colledge of Science and Technology, Nihon University, Narashino-dai, Funabashi 274-8501, Japan 2 Faculty of Science and Technology, Keio University, Hiyoshi, Yokohama 223-8522, Japan

Received  September 2014 Revised  February 2015 Published  November 2015

We consider limiting behavior of multi-dimensional diffusion processes in two types of Brownian environments. One is given values at different $d$ points of a one-dimensional Brownian motion, which is supposed to be a multi-parameter environment, and other is given by $d$ independent one-dimensional Brownian motions. We show recurrence of multi-dimensional diffusion processes in both Brownian environments above for any dimension and almost all environments. Their limiting behavior is quite different from that of ordinary multi-dimensional Brownian motion. We also consider cases of reflected Brownian environments.
Citation: Hiroshi Takahashi, Yozo Tamura. Recurrence of multi-dimensional diffusion processes in Brownian environments. Conference Publications, 2015, 2015 (special) : 1034-1040. doi: 10.3934/proc.2015.1034
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